反射偏微分方程解的惩罚近似  

A Penalty Approximation Method for Reflected Partial Differential Equations

作　　者：马治山 杜巩胜 

机构地区：[1]北京邮电大学理学院,北京

出　　处：《理论数学》2024年第2期719-732,共14页Pure Mathematics

摘　　要：本文研究了一类二阶非线性抛物型偏微分方程反射问题,通过构造反射项的近似形式,给出了此反射方程的近似方程,即反射偏微分方程的惩罚方程。为了弱化条件,我们引入与反射问题等价的变分不等式问题,证明了惩罚方程的解收敛于变分不等式问题的解,此外由于反射项近似的特殊性,我们得到了其收敛速度,并且可以通过调整惩罚方程中的参数来控制反射方程解的精度。In this article, we mainly study a class of second-order nonlinear parabolic partial differential equation with one reflecting wall. By constructing an approximate form of the reflection term, we provide an approximate equation for this reflection equation, which is the penalty equation for the reflection partial differential equation. In order to weaken the condition, we introduce a variational inequality problem equivalent to the reflection problem and prove that the solution of the penalty equation converges to the solution of the variational inequality problem. Furthermore, due to the particularity of the approximate reflection term, the convergence rate of the solution is given. And the accuracy of the reflection equation solution can be controlled by adjusting the parameters in the penalty equation.

关 键 词：偏微分方程 反射问题 惩罚方法 变分不等式 

分 类 号：O17[理学—数学]